Extensions

Applications

This entry is about the notion of adjoint triple involving three functors. This is not to be confused with the notion of adjoint monads, which were also sometimes called adjoint triples, with “triple” then being a synonym for “monad”. However, an adjoint triple in the sense here does induce an adjoint monad!

In general there is a duality (an antiequivalence of categories) between the category of monads having right adjoints and comonads having left adjoints. Note also that the algebras for a left-adjoint monad can be identified with the coalgebras for its right adjoint comonad. (Theorems 5.8.1 and 5.8.2 in (SGL).)

Fully faithful adjoint triples

Proposition

For an adjoint triple F⊣G⊣HF\dashv G\dashv H we have that FF is fully faithful precisely if HH is fully faithful.

The preceeding proposition is folklore; perhaps its earliest appearance in print is (DT, Lemma 1.3). A slightly shorter proof is in (KL, Prop. 2.3). Both proofs explicitly exhibit an inverse to the counit GH→IdG H \to Id or the unit Id→GFId \to G F given an inverse to the other (which could be extracted by beta-reducing the above, slightly more abstract argument). It also appears in (SGL, Lemma 7.4.1).

In the situation of Proposition 2, we say that F⊣G⊣HF\dashv G \dashv H is a fully faithful adjoint triple. This is often the case when DD is a category of “spaces” structured over CC, where FF and HH construct “discrete” and “codiscrete” spaces respectively.

For instance, if G:D→CG\colon D\to C is a topological concrete category, then it has both a left and right adjoint which are fully faithful. Not every fully faithful adjoint triple is a topological concrete category (among other things, GG need not be faithful), but they do exhibit certain similar phenomena. In particular, we have the following.

Proposition

Suppose (F⊣G⊣H):C→D(F \dashv G \dashv H) \colon C \to D is an adjoint triple in which FF and HH are fully faithful, and suppose that CC is cocomplete. Then GG admits final lifts for smallGG-structured sinks.

Proof

Let {G(Si)→X}\{G(S_i) \to X\} be a small sink in CC, and consider the diagram in DD consisting of all the SiS_i, all the counits ε:FG(Si)→Si\varepsilon\colon F G(S_i) \to S_i (where FF is the left adjoint of GG), and all the images FG(Si)→F(X)F G(S_i) \to F(X) of the morphisms making up the sink. The colimit of this diagram is preserved by GG (since it has a right adjoint as well). But the image of the diagram consists essentially of just the sink itself (since FF is fully faithful, G(ε)G(\varepsilon) is an isomorphism), and its colimit is XX; hence the colimit of the original diagram is a lifting of XX to DD (up to isomorphism). It is easy to verify that this lifting has the correct universal property.

Idempotent adjoint triples

Proposition

For an adjoint triple F⊣G⊣HF\dashv G\dashv H, the adjunction F⊣GF\dashv G is an idempotent adjunction if and only if the adjunction G⊣HG\dashv H is so.

Proof

The monad GFG F is left adjoint to the comonad HGH G, with the structure maps being mates. Therefore, by a standard fact, the category of GFG F-algebras and the category of HGH G-coalgebras are isomorphic over their common base. However, F⊣GF\dashv G is idempotent precisely when GFG F is an idempotent monad, hence precisely when the forgetful functor of the category of GFG F-algebras is fully faithful, and dually for G⊣HG\dashv H. Since the categories of algebras are isomorphic respecting their forgetful functors, one forgetful functor is fully faithful if and only if the other is.

Specific examples

Given any ringhomomorphismf∘:R→Sf^\circ: R\to S (in commutative case dual to an affine morphismf:SpecS→SpecRf: Spec S\to Spec R of affine schemes), there is an adjoint triple f!⊣f*⊣f*f^!\dashv f_*\dashv f^* where f*:RMod→SModf^*: {}_R Mod\to {}_S Mod is an extension of scalars, f*:SMod→RModf_*: {}_S Mod\to {}_R Mod the restriction of scalars and f!:M↦HomR(RS,RM)f^! : M\mapsto Hom_R ({}_R S, {}_R M) its right adjoint. This triple is affine in the above sense.

If TT is a lax-idempotent 2-monad, then a TT-algebra AA has an adjunction a:TA⇄A:ηAa : T A \rightleftarrows A : \eta_A. If this extends to an adjoint triple with a further left adjoint to aa, then AA is called a continuous algebra.